Hence, it suffices to show that (1) \(G^{\prime }\) 
is 2-connected and (2) \(G^{\prime }\)  has fewer
separating 2-sets than \(G\) . The assertion (1) follows from the “ear” decomposition characterization, e.g., [1]} (p. 163). One can form \(G^{\prime }\)  by adding paths
(ears) in the order \(v_1, y_1, w_1, vy_1, v_2, y_2, w_2, \ldots , vy_n\) .
